The Recovery Phrase is the single secret piece of data that is used to generate both the public and private key for your account.
LOBSTR Vault uses the Recovery Phrase instead of the private key for your convenience.
Your private key is still stored locally and used to encrypt data and sign transactions.
LOBSTR Vault is not a wallet app and does not expose the private key for security reasons.
If you need the private key of your LOBSTR Vault account, you would need to derive it from your 12-word Recovery Phrase using third-party tools.
If you are using the Vault Signer Card, the secret key of your signer account is stored on the highly-secure chip embedded into the Signer Card. It never leaves the chip and is never sent to any servers. It is impossible to clone, derive or export the signer key, even for the manufacturer of the card, as audited by Kudelski Security (JP Aumasson)